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\author{五六七 }
\title{用差分法求解拉普拉斯方程 }

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\begin{abstract}
使用差分法求解一个偏微分方程的边值问题。
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\section{问题描述}
考虑如下的拉普拉斯方程的第一边值问题
\begin{eqnarray}
\left\{\begin{array}{l}
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \,\, (x,y)\in \Omega, \\  
u(x,y)\mid_{(x,y)\in \partial \Gamma} = \ln((1+x)^2+y^2). 
\end{array}\right. 
\end{eqnarray}
其中 $\Omega = \{(x,y)\mid 0\le x,y\le 1\}$. 取 $h=\tau=\frac{1}{3}$. 

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\section{建立模型}




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\section{编程计算}




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\section{回答问题}




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%\section{参考文献 }
\begin{thebibliography}{99}

%\bibitem{dingtongren} 丁同仁、李承治，常微分方程教程，高等教育出版社，2022年3月第三版。
\bibitem{sishoukui-2} 司守奎,孙玺菁. \emph{Python数学建模算法与应用}, 国防工业出版社. 2022年1月第1版. 
%\bibitem{hexiaoqun-ara} 何晓群. \emph{应用回归分析(R语言版)}. 电子工业出版社. 2017年7月第1版. 
%\bibitem{dalgaard} Peter Dalgaard 著, 郝智恒等译. \emph{R语言统计入门}. 人民邮电出版社. 2014年6月第1版. 


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